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\begin{document}

{\bf Math 107\ \ \  Fall '12 \quad\quad\quad Exam 1 \hfill
\makebox[1.5 in]{Score: \hrulefill}\\

\makebox[3 in]{Name: \hrulefill} \hfill \makebox[3 in]{TA's Name:
\hrulefill}}
\medskip \hrule\smallskip \noindent {\bf
Instructions:} You must show supporting work to receive full and
partial credits. No text book, notes, formula sheets allowed.
\smallskip\hrule
%\begin{center}
%\begin{tabular}[t]{|c|c|c|c|c|c|c|c|}
%\hline No.&\quad 1 \quad & \quad 2 \quad & \quad 3\quad &\quad
%4\quad & \quad 5 \quad &\quad 6\quad &
%\quad Total \quad\\ \hline score & & & & & & & \\
%\hline
%\end{tabular}
%\end{center}
\begin{enumerate}
\item[\bf 1(13pts)] Use the definition to determine if
the improper integral converge: $\displaystyle{\int_1^2
\frac{1}{x-1} dx}$. ({\bf  Show all work.})

\vskip 4in

\item[\bf 2(12pts)] %Use integration by parts to find
%$\displaystyle{\int x\ln x\, dx}$
Some values of a function $y=f(x)$ are given in the table below.
\begin{center}
\vskip -.2in
\begin{tabular}[t]{|c|c|c|c|c|c|}
\hline $x$ &\quad 1 \quad & \quad 1.5 \quad & \quad 2\quad &\quad
2.5\quad & \quad 3 \quad \\ \hline $f(x)$ &\quad 1 \quad &\quad 2.25
\quad &\quad 4 \quad
&\quad 6.25 \quad &\quad 9 \quad \\
\hline
\end{tabular}
\end{center}

(a) Use the right point sum with two partition, $R_2$, to
approximate $\displaystyle \int_1^3f(x)dx$.

\vskip 1.25in

(b) Use the midpoint sum, $M_2$, to approximate $\displaystyle
\int_1^3f(x)dx$.

\vskip 1in






\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak


\item[\bf 3(15pts)] Let $R$ be the region in the first quadrant that is
bounded by two curves: $y=x^3$ and $y=x$. Sketch the region and set
up a definite integral for its area. \textbf{Do not evaluate the
integral.}



%\vspace{2.2in}

%\noindent (b) Use the method of washer to set up an integral whose
%value gives the volume of the solid obtained by revolving the region
%$R$ about the vertical line $x=1$. {\bf Do not evaluate the
%integral.}

\vspace{3.5in}

%\noindent (c)

\item[\bf 4(10pts)] The base of a solid is a quarter disk bounded by
$x^2+y^2=1,\ x\ge 0, y\ge 0$. Each cross section of the solid that
is perpendicular to the $x$-axis is a square as shown. Set up an
integral for the volume of the solid. {\bf Do not evaluate the
integral.}

\rightskip 0in
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\vspace{2.5in}

\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak

\item[\bf 5(10pts)] Let $r=f(\theta)=\theta$ be the Archimedes
spiral. Sketch the curve for $0\le\theta\le\pi/2$ and set up an
integral for the area of the region that is bounded by the spiral
between $\theta=0$ and $\theta=\pi/2$. \textbf{Do not evaluate the
integral.}

%Find the partial fraction for the rational function
%$\displaystyle{\frac{x+1}{(1-x)(x^2+1)}}$.


\vspace{3.5in}

\item[\bf 6(15pts)] A doughnut solid is obtained by rotating the unit
disk bounded by $x^2+y^2=1$ around the vertical line $x=2$. Set up a
definite integral for the volume of the doughnut. \textbf{Do not
evaluate the integral.}


\vspace {3.2 in}

\vfill \hfill \makebox[1.5 in]{(\it Page \arabic{page}. Continue on
Next Page ... \rm )} \pagebreak

\item[\bf 7(15pts)] Set up an integral for the length of the
curve $\displaystyle{y=x^2}$ in the interval $0\le x\le 1$. {\bf Do
not evaluate the integral.}

\vskip 3.5in

\item[\bf 8(10pts)] Use the Comparison Test to determine whether
the improper integral converge or diverge: (Hint: Use $x\ge \sqrt x$
for $x\ge 1$. {\bf Show all work.})
$$\displaystyle{\int_2^\infty \frac{x-\sqrt x}{x^2-1}\, dx}$$.


\vskip 2.5 in

\end{enumerate}

\vfill \noindent \textbf{2 point Bonus Question}: The first day of
this year's Fall is $\underline{\ \ \ \ \ \ \ \ \ \ }$. \hrule
\hfill \makebox[1.5 in]{(\it Page \arabic{page}. \hskip 1in ... The
End\rm )}
\end{document}
